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Categorical Quantum Dynamics Categorical Quantum Dynamics Stefano Gogioso Trinity College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Michaelmas 2016 Per Aspera Ad Astra Contents 1 Introduction 1 1.1 Summary of this work . .1 1.2 Background literature . .2 1.2.1 Categorical Quantum Mechanics . .2 1.2.2 Dagger compact structure . .3 1.2.3 Some application of CQM . .3 1.2.4 The ZX calculus . .4 1.2.5 Applications of strong complementarity . .4 1.3 Brief synopsis of this work . .6 1.3.1 A primer of CQM . .6 1.3.2 Coherent dynamics and symmetries . .6 1.3.3 Strong complementarity in quantum algorithms . .7 2 Categorical Quantum Mechanics 8 2.1 Symmetric monoidal categories . .8 2.1.1 Objects as physical systems . .8 2.1.2 Sequential composition of processes . .8 2.1.3 Parallel composition of processes . .9 2.1.4 States, effects and scalars . 11 2.1.5 Examples of symmetric monoidal categories . 12 2.2 Dagger-compact categories . 14 2.2.1 Dagger, isometries and unitaries . 14 2.2.2 Dagger compact structure . 15 2.2.3 Examples of dagger compact categories . 17 2.2.4 The matrix algebra . 19 2.3 Environments, causality and purification . 19 2.3.1 Environment structures . 19 2.3.2 Discarding maps in categories of matrices . 20 i 2.3.3 The CPM construction . 21 2.3.4 Purification . 24 2.4 Coherent data manipulation . 26 2.4.1 Dagger Frobenius algebras . 26 2.4.2 Quantum observables . 27 2.4.3 A brief digression on observables . 29 2.4.4 Coherent data manipulation . 31 2.4.5 Bell states and effects . 33 2.4.6 Observables in classical physics . 33 2.4.7 Canonical Frobenius algebras . 35 2.5 Measurements, decoherence and classicality . 35 2.5.1 Probabilistic theories . 35 2.6 The CP* construction . 37 2.6.1 The CP* construction and quantum theory . 37 2.6.2 The CP* construction and R-probabilistic theories . 39 2.7 Non-locality and contextuality . 41 2.7.1 The sheaf-theoretic framework . 41 2.7.2 Contextuality in R-probabilistic theories . 47 2.8 Some toy models of quantum theory . 49 2.8.1 Theories of wavefunctions over semirings . 49 2.8.2 Real quantum theory . 52 2.8.3 Relational quantum theory . 53 2.8.4 Hyperbolic quantum theory . 53 2.8.5 Parity quantum theory . 54 2.8.6 Finite-field quantum theory . 56 2.8.7 p-adic quantum theory . 57 2.8.8 Tropical quantum theory . 58 3 Categorical Quantum Dynamics 61 3.1 Introduction . 61 3.1.1 A coherent approach to quantum symmetries . 61 3.1.2 Synopsis of this Chapter . 62 3.2 Coherent Groups . 68 3.2.1 Wavefunctions on periodic lattices . 68 3.2.2 Complementarity and strong complementarity . 73 3.2.3 Coherent Groups . 77 ii 3.3 Wavefunctions on a periodic lattice . 85 3.3.1 What we look for in a momentum observable . 85 3.3.2 Momenta generate translation symmetry . 89 3.3.3 Positions generate boost symmetry . 92 3.3.4 Weyl Canonical Commutation Relations . 99 3.3.5 Uncertainty principle . 101 3.4 Systems with symmetries . 104 3.4.1 Unitary representations of coherent groups . 104 3.4.2 The category of representations of a coherent group . 107 3.4.3 Symmetry-observable duality . 110 3.4.4 Stone's theorem revisited . 119 3.5 Infinite-dimensional CQM . 124 3.5.1 Non-standard analysis . 125 3.5.2 The category ?Hilb . 132 3.5.3 How to use ?Hilb for standard purposes . 139 3.5.4 Infinite-dimensional categorical quantum mechanics . 142 3.5.5 Wavefunctions with periodic boundaries . 144 3.6 Quantum dynamics . 150 3.6.1 A traditional perspective on quantum dynamics . 150 3.6.2 Quantum clocks . 160 3.6.3 Quantum dynamical systems . 160 3.6.4 The coherent Hamiltonian . 162 3.6.5 Schr¨odinger's Equation . 163 3.6.6 von Neumann's mean ergodic theorem . 165 3.6.7 Stone's Theorem . 167 3.6.8 Feynman's clock . 168 3.6.9 Clock-system synchronisation . 173 3.6.10 Time observables . 177 4 Strong Complementarity in Quantum Algorithms 189 4.1 Hidden Subgroup Problem . 190 4.1.1 The Hidden Subgroup Problem . 191 4.1.2 The Quantum Algorithm . 192 4.1.3 Non-abelian HSP . 198 4.1.4 Toy quantum theories . 199 4.2 Mermin-type non-locality scenarios . 204 iii 4.2.1 Mermin's Original Argument . 206 4.2.2 The phase group . 210 4.2.3 Generalised Mermin-type Arguments . 218 4.2.4 Quantum realisability . 226 4.2.5 All-vs-Nothing Arguments . 228 4.2.6 A fully worked-out example . 231 4.2.7 Quantum-classical Secret Sharing . 236 Bibliography 253 iv Chapter 1 Introduction 1.1 Summary of this work Since its original introduction, strongly complementary observables have been a fundamental ingredient of the ZX calculus, one of the most successful fragments of Categorical Quantum Mechanics (CQM). In this thesis, we show that strong complementarity plays a vastly greater role in quantum theory. Firstly, we use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Secondly, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and prove that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin- type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future. 1 1.2 Background literature 1.2.1 Categorical Quantum Mechanics This work takes its roots in the framework of categorical quantum mechanics (CQM) [AC09, CK15], which its extends and refines in a number of aspects. The general motivation behind the application of category-theoretic tools lies in the intuition that the features distinguishing quantum theory from classical physics can be understood in terms of the way quantum processes compose, sequentially and in parallel (we refer to this as an operational, or process-theoretic, description of quantum theory). The framework of symmetric monoidal categories is particularly suited for operational descriptions, and comes with a natural diagrammatic formalism [JS91, BS10, CK15, Kis12, Sel07, CH12], combining the rigour of the category theory with the versatility of graphical manipulation. The diagrammatic formalism has quickly become a major selling point of CQM, and receives plenty of interest on its own [CD11, Bac14, Had15, Kis12, KZ15]. The operational and process-theoretic description of quantum theory given by CQM [AH12b, CK15] is in a certain sense antipodal to the traditional Hilbert space formulation, which heavily relies on explicit complex-linear structure. Several interme- diate approaches exist, such as quantum circuits and generalised probabilistic theories [Har01, Bar07], and have been applied to a diversity of topics in quantum information and foundations. The framework of operational probabilistic theories (OPTs) is perhaps the closest, in spirit, to CQM: it was developed in [CDP10, CDP11, CS15] with the aim of obtaining an informational derivation of quantum theory, it comes with a graphical calculus and it has a strong operational and process-theoretic flavour to it; work to connect OPTs to CQM is currently being undertaken [Tul16, GS16]. Despite the diagrammatic and operational similarities, the approach to quantum theory of OPTs is quite different from that of CQM: the former relies on explicit probabilistic structure, and is mainly axiomatic in nature, while the latter focuses on categorical structures (with no explicit summations or probabilities), and is mostly constructive. The processes which OPTs are concerned with are physical processes1, or processes appearing in their convex decompositions2. On the contrary, CQM is concerned with the study of arbitrary processes with interesting categorical properties, which are used as building blocks of physical processes. In a nutshell, OPTs take physical processes and impose axioms on the way they can be probabilistically decomposed, while CQM 1Such as quantum channels, aka completely positive trace-preserving maps. 2Such as sub-normalised pure states, appearing in the convex decomposition of density matrices. 2 studies how certain building blocks compose and interact to form larger processes of physical interest. 1.2.2 Dagger compact structure The categorical environment of choice for pure-state quantum theory in CQM is that of dagger compact symmetric monoidal categories: the dagger structure is the categorical abstraction of operator adjunction3 in the Hilbert space formalism, while the compact structure corresponds to operator-state duality. Within this environment, the most common building blocks used by CQM are certainly y-Frobenius algebras, which provide the coherent/pure versions of the classical operations of copy, delete and match, and are a key component in the treatment of classicality [CP07, CPP10].
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